# Properties

 Label 193200ef Number of curves $6$ Conductor $193200$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("193200.df1")

sage: E.isogeny_class()

## Elliptic curves in class 193200ef

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
193200.df5 193200ef1 [0, -1, 0, 50392, -8934288] [2] 1572864 $$\Gamma_0(N)$$-optimal
193200.df4 193200ef2 [0, -1, 0, -461608, -103142288] [2, 2] 3145728
193200.df3 193200ef3 [0, -1, 0, -2029608, 1013273712] [2, 2] 6291456
193200.df2 193200ef4 [0, -1, 0, -7085608, -7257062288] [2] 6291456
193200.df1 193200ef5 [0, -1, 0, -31653608, 68555993712] [2] 12582912
193200.df6 193200ef6 [0, -1, 0, 2506392, 4896089712] [2] 12582912

## Rank

sage: E.rank()

The elliptic curves in class 193200ef have rank $$1$$.

## Modular form 193200.2.a.df

sage: E.q_eigenform(10)

$$q - q^{3} + q^{7} + q^{9} + 4q^{11} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.